EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIECULAR SECTION. 925 



(p', w\ z'), and in the second case the solution given for an element (Pq, ^o) ^o) t)f 

 traction at {a, w\ z'). 



Denote the displacements of the three systems, in the order in which they are 

 mentioned, by u^, u^, ii, ; itj, uj, uj ; t?/', u^' , u" . 



Let the part of the solid to which Betti's Theorem is applied be bounded by the 

 two planes 2 = z^ and z = 2.2, where Zi<z' or 2" and Z:^z' or z". 



From the facts that only transitory modes vanishing at infinity, and permanent 

 terms which are the same except for sign on the two sides of the source, occur in our 

 solutions, it follows that the contributions to the work expressions from the two end- 

 planes neutralise each other, as may easily be seen by supposing them to be taken at 

 a very great distance. 



Thus we get simply 



Pw/(p", w", 2") + n?/„'(p", w", 2") + 7,U,'{P'> <"'\ Z") = PoWp(p'. '^'. 2') + ^o'«a,(p'. w', Z) + Z,M,(P', w', Z'), (1) 



and 



PUp"{p", oj", z") +UuJ'(p", (1)', z") + Zu"{p", 0)", z") = FgUp{a, <»', z) +U„u^{a, 0/, z') + Z^)U^{a, co', z'). (2) 



It is obvious that the right-hand member of (1) passes continuously into the right- 

 hand member of (2) as f)' approaches a ; hence the same must be true of the left-hand 

 members. Thus, since P, O, Z are arbitrary, the values of uJ, uJ, ul at any point 

 pass continuously into the values of u^', uJ', ul' there as p approaches a ; which is 

 the theorem. 



It is easy to see that the limits for p' = a of the solutions given above for a force at 

 {p, u)', z') are found by simply putting p' = a in every term. Hence we need no longer 

 distinguish between a unit force and a unit element of surface traction, but may refer 

 to the latter simply as a unit force at (a, w', z'). 



21. The six permanent straining modes, or St-Venaiit's solutions. 



We give here a table of the displacements and the most important stresses in 

 rectangular and cylindrical coordinates for the six permanent straining modes {cf. 

 Art. 3). 



It is convenient from this point to introduce, instead of the symbol f = ■ _^j ^ 



X + ^i 



the symbols in common use : 



o- = Poisson's ratio = 



2(A + /x)' 

 and E = Young's modulus = ^ — = 2/a(cr + 1 ). 



(1) 



The following relations may be useful for reference : — 



, = 4-4<r,, i-p' = (r, ':^=-(c. + |-), /x(8-v) = 2K ... (2) 



We shall also use a symbol r defined by the equations 



E v^-20v + 92 1/E /.\ ■ 



"~^- 3(8-vy^ =yl7. + ^~E> • • ■ • (3) 



